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Decomposition of \( S_{12}^{\mathrm{new}}(13) \) into irreducible Hecke orbits

magma: S := CuspForms(13,12);
magma: N := Newforms(S);
sage: N = Newforms(13,12,names="a")
Label Dimension Field $q$-expansion of eigenform
13.12.1.a 5 $\Q(\alpha_{ 1 })$ \(q \) \(\mathstrut+\) \(\alpha_{1} q^{2} \) \(\mathstrut+\) \(\bigl(\frac{1}{24096} \alpha_{1} ^{4} \) \(\mathstrut+ \frac{13}{2008} \alpha_{1} ^{3} \) \(\mathstrut- \frac{8167}{24096} \alpha_{1} ^{2} \) \(\mathstrut- \frac{93171}{4016} \alpha_{1} \) \(\mathstrut+ \frac{330631}{502}\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{1} ^{2} \) \(\mathstrut- 2048\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(- \frac{31}{16064} \alpha_{1} ^{4} \) \(\mathstrut- \frac{3581}{16064} \alpha_{1} ^{3} \) \(\mathstrut+ \frac{56183}{8032} \alpha_{1} ^{2} \) \(\mathstrut+ \frac{646467}{1004} \alpha_{1} \) \(\mathstrut- \frac{2601597}{251}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(\frac{115}{24096} \alpha_{1} ^{4} \) \(\mathstrut- \frac{797}{8032} \alpha_{1} ^{3} \) \(\mathstrut- \frac{210947}{12048} \alpha_{1} ^{2} \) \(\mathstrut+ \frac{150195}{502} \alpha_{1} \) \(\mathstrut- \frac{86680}{251}\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(- \frac{85}{8032} \alpha_{1} ^{4} \) \(\mathstrut+ \frac{3159}{4016} \alpha_{1} ^{3} \) \(\mathstrut+ \frac{463777}{8032} \alpha_{1} ^{2} \) \(\mathstrut- \frac{13790493}{4016} \alpha_{1} \) \(\mathstrut- \frac{10975231}{502}\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(\alpha_{1} ^{3} \) \(\mathstrut- 4096 \alpha_{1} \bigr)q^{8} \) \(\mathstrut+\) \(\bigl(\frac{4393}{48192} \alpha_{1} ^{4} \) \(\mathstrut+ \frac{3289}{16064} \alpha_{1} ^{3} \) \(\mathstrut- \frac{10638857}{24096} \alpha_{1} ^{2} \) \(\mathstrut+ \frac{326391}{1004} \alpha_{1} \) \(\mathstrut+ \frac{71874862}{251}\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)
13.12.1.b 6 $\Q(\alpha_{ 2 })$ \(q \) \(\mathstrut+\) \(\alpha_{2} q^{2} \) \(\mathstrut+\) \(\bigl(\frac{1003}{200877312} \alpha_{2} ^{5} \) \(\mathstrut+ \frac{156023}{1004386560} \alpha_{2} ^{4} \) \(\mathstrut- \frac{9091847}{167397760} \alpha_{2} ^{3} \) \(\mathstrut- \frac{22495961}{20924720} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{8150117921}{62774160} \alpha_{2} \) \(\mathstrut+ \frac{416654549}{1569354}\bigr)q^{3} \) \(\mathstrut+\) \(\bigl(\alpha_{2} ^{2} \) \(\mathstrut- 2048\bigr)q^{4} \) \(\mathstrut+\) \(\bigl(- \frac{2749}{83698880} \alpha_{2} ^{5} \) \(\mathstrut+ \frac{28591}{83698880} \alpha_{2} ^{4} \) \(\mathstrut+ \frac{10220437}{41849440} \alpha_{2} ^{3} \) \(\mathstrut- \frac{1794561}{10462360} \alpha_{2} ^{2} \) \(\mathstrut- \frac{197366715}{1046236} \alpha_{2} \) \(\mathstrut- \frac{1236090445}{261559}\bigr)q^{5} \) \(\mathstrut+\) \(\bigl(\frac{53981}{125548320} \alpha_{2} ^{5} \) \(\mathstrut+ \frac{882901}{125548320} \alpha_{2} ^{4} \) \(\mathstrut- \frac{85524481}{20924720} \alpha_{2} ^{3} \) \(\mathstrut- \frac{174826841}{2615590} \alpha_{2} ^{2} \) \(\mathstrut+ \frac{12911252227}{1569354} \alpha_{2} \) \(\mathstrut+ \frac{44351075260}{784677}\bigr)q^{6} \) \(\mathstrut+\) \(\bigl(- \frac{186079}{334795520} \alpha_{2} ^{5} \) \(\mathstrut- \frac{4155231}{334795520} \alpha_{2} ^{4} \) \(\mathstrut+ \frac{861069021}{167397760} \alpha_{2} ^{3} \) \(\mathstrut+ \frac{2025228199}{20924720} \alpha_{2} ^{2} \) \(\mathstrut- \frac{191024723449}{20924720} \alpha_{2} \) \(\mathstrut- \frac{23579319285}{523118}\bigr)q^{7} \) \(\mathstrut+\) \(\bigl(\alpha_{2} ^{3} \) \(\mathstrut- 4096 \alpha_{2} \bigr)q^{8} \) \(\mathstrut+\) \(\bigl(- \frac{372733}{251096640} \alpha_{2} ^{5} \) \(\mathstrut+ \frac{3551591}{251096640} \alpha_{2} ^{4} \) \(\mathstrut+ \frac{525938827}{41849440} \alpha_{2} ^{3} \) \(\mathstrut- \frac{55241729}{2092472} \alpha_{2} ^{2} \) \(\mathstrut- \frac{307242655327}{15693540} \alpha_{2} \) \(\mathstrut+ \frac{26120870500}{784677}\bigr)q^{9} \) \(\mathstrut+O(q^{10}) \)

The coefficient fields are:

Coefficient field Minimal polynomial of $\alpha_j$ over $\Q$
$\Q(\alpha_{ 1 })$ \(x ^{5} \) \(\mathstrut +\mathstrut 41 x ^{4} \) \(\mathstrut -\mathstrut 5776 x ^{3} \) \(\mathstrut -\mathstrut 137132 x ^{2} \) \(\mathstrut +\mathstrut 8660928 x \) \(\mathstrut +\mathstrut 8321280\)
$\Q(\alpha_{ 2 })$ \(x ^{6} \) \(\mathstrut -\mathstrut 55 x ^{5} \) \(\mathstrut -\mathstrut 12286 x ^{4} \) \(\mathstrut +\mathstrut 603264 x ^{3} \) \(\mathstrut +\mathstrut 39388912 x ^{2} \) \(\mathstrut -\mathstrut 1594524928 x \) \(\mathstrut -\mathstrut 11319915520\)

Decomposition of \( S_{12}^{\mathrm{old}}(13) \) into lower level spaces

\( S_{12}^{\mathrm{old}}(13) \) \(\cong\) $ \href{ /ModularForm/GL2/Q/holomorphic/1/12/1/ }{ S^{ new }_{ 12 }(\Gamma_0(1)) }^{\oplus 2 } $